#7683395

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(6,9)(10,11)(14,19)(17,28)(20,26)(21,23);(1,2)(3,4)(13,14,18,19)(15,20,24,26)(16,23,27,21)(17,22,28,25);(5,6,8,9)(7,11,12,10)(14,19)(17,28)(20,26)(21,23);(5,7,8,12)(6,10,9,11)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(13,16,18,27)(14,21,19,23)(15,22,24,25)(17,26,28,20);(13,17)(14,22)(15,23)(16,26)(18,28)(19,25)(20,27)(21,24)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590591);